I'm sorry for your loss
I think having her tested is a good idea.
Yes, put her in the fridge, but don't freeze.
I'm sorry she is declining.
Were you able to get any vitamins for her? Is she able to eat and drink?
I assume you have her separated.
You might want to look at making her a sling so she is a bit more upright.
It sounds like she could smother if she continues having her face pushed down in the bedding.
As far as necropsy reports go, I am not sure what information you will receive. Eggcessive or Casportpony are more knowledge on that subject.